Trisecting an Angle Using the Conchoid of Nicomedes

Date: 08/16/2002 at 20:30:39
From: J. A. Greene
Subject: Trisecting an angle and using a Nichomedian choncoid

I attempted to trisect an angle using the Nichodemian Choncoid (which 
I constructed using a compass and straightedge), and I showed it to my 
dad, who said that it was 100% impossible to do so, and I showed it to 
my mom, who said that I might actually have trisected the angle.  I 
want to know if it possible that I could have trisected it using the 
conchoid.

JA Greene

Date: 08/16/2002 at 23:13:37
From: Doctor Peterson
Subject: Re: Trisecting an angle and using a Nichomedian choncoid

Hi, J. A.

Of course it is possible! The Greeks invented this curve for the 
purpose of trisecting angles; it's nothing new. The problem is that it 
doesn't follow the rules for a compass and straightedge construction, 
which must use NOTHING else. That is what is impossible, not 
trisection using other tools. You didn't really construct the curve 
with compass and straightedge alone, since it is not composed only of 
lines and circles; you must have constructed some points and then 
drawn a curve through them somehow.

See the Dr. Math FAQ:

   Impossible constructions
   http://mathforum.org/dr.math/faq/faq.impossible.construct.html 

and this article from the MacTutor History of Mathematics archive at 
St. Andrews University::

   Trisecting an angle
   <http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/
Trisecting_an_angle.html>

And see this on the conchoid:

   Conchoid of Nicomedes - Xah Lee
   <http://www.xahlee.org/SpecialPlaneCurves_dir/
ConchoidOfNicomedes_dir/conchoidOfNicomedes.html>

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/